LINEAR ALGEBRA

Example : Find x1 and x2 from these equation : Solution :

General form : Where : a1, a2, a3 .. an and b are constantas x1, x2, x3 .. xn are variables

Linear Equation in matrix form If we have some equations : Then, we can write :

General form : Where : Or, we can write :

Example : Find x1 and x2 from these equation : Solution : Find A-1 ....

A-1 : Formula :

Cramer’s rules Assume : Determinants :

x1 : x2 : x3 :

Example : Find x1 and x2 from these equation : Solution :

x1 and x2 : Proof :

Find x1, x2 and x3 from these equations :

Find Determinants :

Find x1, x2 and x3 :

Adjoint Matrix Adjoint matrix of a square matrix is the transpose of the matrix formed by cofactors of elements of determinant |A| How to calculate adjoint : Calculate minor matrix for each element of matrix Make cofactor matrix cofactor is a sign minor, denoted by : Cij = (-1)ij . Mij Change to Transpose matrix.

Example Find inverse for A : Calculate |A| : =(1.5.3 + 2.0.2 + 3.0.4) – (3.5.2 +1.0.4 + 2.0.3) = 15 – 30 = - 15

Make a new matrix with minor and cofactor → Transpose that matrix :

Find x1, x2 and x3 from these equations :

Matrix form : Formula :

Determinants : Use minor cofactor :

New matrix K : A-1 :

Formula :

Questions x + y + z – 6 = 0 2x – z + 1 = 0 x – y + 2z – 5 = 0 Find x1 and x2 from these equations : 2x1 + x2 – 4 = 0 x1 – 3x2 + 5 = 0 Find x, y and z from these equations : x + y + z – 6 = 0 2x – z + 1 = 0 x – y + 2z – 5 = 0